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the factor theorem shows that one has a factorization = (), where both factors have integer coefficients (the fact that Q has integer coefficients results from the above formula for the quotient of P(x) by /).
This factorization is also unique up to the choice of a sign. For example, + + + = + + + is a factorization into content and primitive part. Gauss proved that the product of two primitive polynomials is also primitive (Gauss's lemma). This implies that a primitive polynomial is irreducible over the rationals if and only if it is irreducible ...
Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division : checking if the number is divisible by prime numbers 2 ...
Any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Fundamental Theorem of Arithmetic is, in fact, a special case of the unique factorization theorem in commutative Möbius monoids.
An LU factorization refers to expression of A into product of two factors ... there is a closed (explicit) formula for the elements of L, D, ...
Visual proof of the formulas for the sum and difference of two cubes In mathematics , the sum of two cubes is a cubed number added to another cubed number. Factorization
Euler's factorization method is a technique for factoring a number by writing it as a sum of two squares in two different ways. For example the number 1000009 {\displaystyle 1000009} can be written as 1000 2 + 3 2 {\displaystyle 1000^{2}+3^{2}} or as 972 2 + 235 2 {\displaystyle 972^{2}+235^{2}} and Euler's method gives the factorization ...
It is clear that any finite set {} of points in the complex plane has an associated polynomial = whose zeroes are precisely at the points of that set. The converse is a consequence of the fundamental theorem of algebra: any polynomial function () in the complex plane has a factorization = (), where a is a non-zero constant and {} is the set of zeroes of ().